Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $r \neq 0$. $n = \dfrac{-2r - 6}{-10r + 10} \div \dfrac{-r - 3}{r^2 - 4r + 3} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{-2r - 6}{-10r + 10} \times \dfrac{r^2 - 4r + 3}{-r - 3} $ First factor the quadratic. $n = \dfrac{-2r - 6}{-10r + 10} \times \dfrac{(r - 1)(r - 3)}{-r - 3} $ Then factor out any other terms. $n = \dfrac{-2(r + 3)}{-10(r - 1)} \times \dfrac{(r - 1)(r - 3)}{-(r + 3)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac{ -2(r + 3) \times (r - 1)(r - 3) } { -10(r - 1) \times -(r + 3) } $ $n = \dfrac{ -2(r + 3)(r - 1)(r - 3)}{ 10(r - 1)(r + 3)} $ Notice that $(r + 3)$ and $(r - 1)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac{ -2\cancel{(r + 3)}(r - 1)(r - 3)}{ 10\cancel{(r - 1)}(r + 3)} $ We are dividing by $r - 1$ , so $r - 1 \neq 0$ Therefore, $r \neq 1$ $n = \dfrac{ -2\cancel{(r + 3)}\cancel{(r - 1)}(r - 3)}{ 10\cancel{(r - 1)}\cancel{(r + 3)}} $ We are dividing by $r + 3$ , so $r + 3 \neq 0$ Therefore, $r \neq -3$ $n = \dfrac{-2(r - 3)}{10} $ $n = \dfrac{-(r - 3)}{5} ; \space r \neq 1 ; \space r \neq -3 $